huber_sf: Huber scoring function

In scoringfunctions: A Collection of Loss Functions for Assessing Point Forecasts

Huber scoring function

Description

The function huber_sf computes the Huber scoring function with parameter a, when y materialises and x is the predictive Huber mean.

The Huber scoring function is defined in Huber (1964).

Usage

huber_sf(x, y, a)

Arguments

x	Predictive Huber mean (prediction). It can be a vector of length n (must have the same length as y).
y	Realisation (true value) of process. It can be a vector of length n (must have the same length as x).
a	It can be a vector of length n (must have the same length as y).

Details

The Huber scoring function is defined by:

S(x, y, a) := \left\{ \begin{array}{ll} \dfrac{1}{2} (x - y)^2, & |x - y| \leq a\\ a |x - y| - \dfrac{1}{2} a^2, & |x - y| > a \end{array} \right.

or equivalently

S(x, y, a) := (1/2) \kappa_{a,a}(x - y) (2 (x - y) - \kappa_{a,a}(x - y))

where \kappa_{a,b}(t) is the capping function defined by:

\kappa_{a,b}(t) := \max \lbrace \min \lbrace t,b \rbrace, -a \rbrace

Domain of function:

x \in \mathbb{R}

y \in \mathbb{R}

a > 0

Range of function:

S(x, y, a) \geq 0, \forall x, y \in \mathbb{R}, a > 0

Value

Vector of Huber losses.

Note

For the definition of Huber mean, see Taggart (2022).

The Huber scoring function is negatively oriented (i.e. the smaller, the better).

The Huber scoring function is strictly \mathbb{F}-consistent for the Huber mean. \mathbb{F} is the family of probability distributions F for which \textnormal{E}_F[Y^2 - (Y - a)^2] and \textnormal{E}_F[Y^2 - (Y + a)^2] (or equivalently \textnormal{E}_F[Y]) exist and are finite (Taggart 2022).

References

Huber PJ (1964) Robust estimation of a location parameter. Annals of Mathematical Statistics 35(1):73–101. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/aoms/1177703732")}.

Taggart RJ (2022) Point forecasting and forecast evaluation with generalized Huber loss. Electronic Journal of Statistics 16:201–231. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/21-EJS1957")}.

Examples

# Compute the Huber scoring function.

df <- data.frame(
    x = c(-3, -2, -1, 0, 1, 2, 3),
    y = c(0, 0, 0, 0, 0, 0, 0),
    a = c(2.7, 2.5, 0.6, 0.7, 0.9, 1.2, 5)
)

df$huber_penalty <- huber_sf(x = df$x, y = df$y, a = df$a)

print(df)

scoringfunctions documentation built on April 4, 2025, 12:28 a.m.